Detecting Anomalies: The Relevance and Power of Standard

Asset Pricing Tests∗

Malcolm Baker, Patrick Luo, Ryan Taliaferro

July 12, 2018

Abstract

The two standard approaches for identifying capital market anomalies are cross-sectionalcoe�cient tests, in the spirit of Fama and MacBeth (1973), and time-series intercept tests, in thespirit of Jensen (1968). A new signal can pass the �rst test, which we label a �score anomaly,�it can pass the second test as a �factor anomaly,� or it can pass both. We demonstrate therelevance of each to a mean-variance optimizing investor facing simple transaction costs that areconstant across stocks. For a risk-neutral investor facing transaction costs, only score anomaliesare relevant. For a risk-averse investor facing no transaction costs, only factor anomalies arerelevant. In the more general case of risk aversion and transaction costs, both tests matter. Inextensions, we derive modi�ed versions of the basic tests that net out anomaly execution costs forsituations where the investor faces capital constraints, a multi-period portfolio choice problem,or transaction costs that vary across stocks. Next, we measure the econometric power of the twotests. The relative power of time-series factor tests falls with the in-sample Sharpe ratio of theincumbent factor model, as in Shanken (1992). New factor anomalies can be successively harderto detect, leading to a lower natural limit on the number of anomalies that can be identi�edby time series tests. Meanwhile, for an investor facing transaction costs, where score anomaliesare also applicable, there can be a higher natural limit on the number of anomalies that can bestatistically validated as relevant.

∗Contact: mbaker@hbs.edu. The authors thank participants at seminars at London Business School and theLondon School of Economics for helpful comments. Please do not quote without permission. The authors would liketo thank Owen Lamont for helpful comments. Malcolm Baker is at the National Bureau of Economic Research andserves as a consultant to Acadian Asset Management, Patrick Luo was previously at Harvard Business School andis a data scientist at Farallon Capital Management, and Ryan Taliaferro is senior vice president at Acadian AssetManagement. In addition, Malcolm Baker and Patrick Luo acknowledge support from the Division of Research atHarvard Business School. The views expressed herein are those of the authors and do not necessarily re�ect the viewsof the National Bureau of Economic Research, Acadian Asset Management, or Farallon Capital Management. Theviews expressed herein should not be considered investment advice and do not constitute or form part of any o�er toissue or sell, or any solicitation of any o�er to subscribe or purchase, shares, units, or other interest in any particularinvestments.

1 Introduction

Capital market anomalies fall into two categories. The �rst is when a security characteristic or

signal, such as the ratio of a �rm's book to market value or its recent change in share price, predicts

future returns but is otherwise not obviously related to risk. The second is when a signal, such as

market beta, is theoretically and empirically connected to portfolio risk but nonetheless does not

predict returns.

Anomalies attract both academic and practical interest. The focus of academic asset pricing

is rationalizing seemingly anomalous predictability, by rede�ning or expanding what an investor

considers to be risk. The focus of academic behavioral �nance is uncovering a parsimonious set of

psychological or sociological biases and institutional frictions that break the standard link between

risk and return. And, the focus of practical investment management is delivering products to

investors with the aim of delivering positive risk-adjusted returns.

In this paper, our two-part goal is to examine the practical relevance and statistical power of

standard asset pricing tests in identifying anomalies. The protagonist we have in mind is an investor

who is seeking to make sensible security selection decisions using historical data as a guide. This

is in the spirit of Brennan, Schwartz, and Lagnado (1997) and Campbell and Viceira (1999), who

examine the consequences of return predictability for portfolio choice in partial equilibrium. While

those papers focus on asset allocation across stocks and bonds, we focus on security selection, much

like the classic analysis of Markowitz (1952) or more recently Garleanu and Pedersen (2013). We

consider a security characteristic or signal to be a relevant anomaly if it has non-zero weight in our

protagonist's selection decision.

The sheer number of potential anomalies accumulated over decades of research demands a certain

degree of simpli�cation. Some recent attempts at grander simpli�cation include Fama and French

(2008, 2015, 2016) and Stambaugh, Yu, and Yuan (2012). Broadly speaking, there are two standard

asset pricing tests. The �rst identi�es candidate anomalies in the �rst category, looking at return

predictability without considering risk. Fama and French (1992) is the canonical citation. The

main empirical tool is cross-sectional return prediction using security-level signals and a pooled

estimation, typically with the procedure of Fama and MacBeth (1973). We call a candidate that

passes this �rst test a �score anomaly.� The second test determines whether a candidate anomaly

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adds to a set of existing return factors. Fama and French (1993) is the canonical citation. The main

empirical tool is an intercept, or alpha, test in a time series return prediction using contemporaneous

factor returns, typically with the procedure of Jensen (1968). We call a candidate that passes this

second test a �factor anomaly.�1

We begin with the question of relevance, and a simple approach, where our protagonist investor

is a Markowitz-style mean-variance optimizer. When our investor is risk averse and faces no trans-

action costs or other frictions, only time-series tests and the factor anomalies that emerge from these

tests are relevant. In that sense, we can think of Fama and French (1993) as catering to the needs of

this type of investor. When our investor is risk neutral and faces a simple form of transaction costs,

constant across securities, only cross-section tests and the score anomalies that emerge are relevant.

Fama and French (1992) are catering to this type of investor. For a risk-averse investor facing simple

transaction costs, both sets of anomalies are relevant, in the sense that our protagonist investor will

not be satis�ed using only those anomalies that emerge from Fama and French (1993) time series

tests. Those anomalies left in the editing room of Fama and French (1992) are also relevant. The

upshot is that academic research might consider either test to be su�cient to establish a new and

relevant asset pricing anomaly.

A caveat is that this framework of relevance ignores practical di�erences across anomalies. In-

vestors face transaction costs that di�er across securities, di�er with capacity constraints, and di�er

in multi-period portfolio choice. Unfortunately, the standard asset pricing tests in their simple form

are no longer relevant, except in special cases. In principle, they can be replaced with intuitive

modi�cations. Essentially, the simple returns and alphas from cross-section and time series tests

can be replaced with returns that are adjusted for execution costs and our investor's speci�c level

of assets under management. More ephemeral anomalies whose conditional score variance is higher

among securities with high transaction costs are, all else equal, less relevant.

Having established relevance, we then turn the power of the two tests. The power of both tests

1A given signal can be both a score anomaly and a factor anomaly. For example, pro�tability is a score anomalyin Table IV of Fama and French (2008) and a factor anomaly in Table 6 of Fama and French (2015). A signal canpass the �rst test as a score anomaly but fail the second. For example, the ratio of book to market value is a scoreanomaly in Table IV of Fama and French (2008) but it is not an independent factor anomaly in Table 6 of Fama andFrench (2015). And, a signal that fails the �rst test can in principle still be a factor anomaly. Members of this groupdo not predict stock-level returns but do hedge contemporaneous returns on other factors. Market beta is a leadingexample. It does not predict returns in the cross section, but it quali�es as a factor anomaly in the sense that it hasa non-zero intercept in factor regressions.

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rises with the number of securities and the number of time periods and falls with idiosyncratic

security variance and factor variance in predictable ways. In addition and importantly, factor tests

have two additional terms. On the one hand, the power of a factor test increases when other factors

are useful in reducing the residual variance of the test factor's time series returns. On the other

hand, the power a factor falls as the in-sample Sharpe ratio of the incumbent factors rises, as in

Shanken (1992). In this sense, there can be a higher chance of a false negative, or Type II error.

Moreover, this tendency rises with the size of the incumbent model, because the in-sample Sharpe

ratio is strictly increasing in the number of incumbent factors. Through this second channel, there is

a lower natural limit to the number of new factor anomalies that can be identi�ed. Intuitively, this

is accentuated in small samples, where degrees of freedom are consumed in parameter estimation.

A �nal note is that the power of time series tests can be resurrected by shortening the return

horizon. This is immediately obvious from the power formula that we derive. Power falls with the

in-sample Sharpe ratio. The Sharpe ratio is itself mechanically increasing in the return horizon,

because returns (the numerator of the ratio) rise linearly with horizon, while standard errors (the

denominator) rise with the square root of horizon. This means asset pricing tests that rely on

quarterly returns have much lower power than tests that rely on monthly horizons, and should

be avoided if possible. And, asset pricing tests with daily return horizons resolve the problem of

relative power: The daily Sharpe ratio is su�ciently small that the di�erence in power between time

series and cross section tests becomes negligible. However, we stop short of recommending that the

standard in asset pricing tests move from monthly to daily. Scholes and Williams (1977)and Liu

and Strong (2008) suggest reasons why inferences might be biased in a daily analysis. The optimal

approach trades o� bias and power. In 30 portfolios from Ken French's data library, analyzing

10-day return horizon essentially solves the problem of bias, suggesting that tests that use longer

return horizons for these anomalies are needlessly sacri�cing power.

We make connections to several related papers along the way. Garleanu and Pedersen (2013)

consider the sort of partial equilibrium analysis that we do here, but they do not consider the

econometric relevance of their optimal portfolios. Instead, they take the return generating process

as given. Moreover, they use one simplifying assumption - that transaction costs are proportional

to risk - to come to an elegant closed-form solution, while we consider a range of less elegant

assumptions about transaction costs. Like us, Hoberg and Welch (2009) consider Fama and French

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(1992) style tests. Their focus, unlike ours, is the use in time series tests of optimized portfolios,

whose returns are derived from cross-sectional regressions, versus sorted portfolios, which are favored

by Fama and French (1993). A large number of papers, including Fama (1998), Mclean and Ponti�

(2016), Harvey, Liu, and Zhu (2015), Bailey and López de Prado (2014), and Novy-marx (2016)

consider the issue of data mining and Type I error, in identifying anomalies that do not really exist.

While this is a serious problem, our focus is instead on power and Type II error, in failing to identify

legitimate anomalies, especially in time series tests. Just as we do, Loughran and Ritter (2000) and

Ang, Liu, and Schwarz (2008) consider the power of asset pricing tests. Loughran and Ritter (2000)

focus on the e�ects of weighting on the power of tests, both in aggregating �rms at a point in time as

well as in aggregating test statistics by market capitalization. Ang, Liu, and Schwarz (2008) focus

on the use of aggregation in estimating factor loadings, while we focus on a comparison of cross-

section and time series tests. We aggregate �rm-level returns into factor returns, and we abstract

from weighting schemes and the estimation of right-hand-side variables - including factor loadings

- in the cross section, and instead focus on the lost power that comes from estimating covariances

in time series tests. Consistent with our logic, Lewellen (2015) �nds strong predictive power in a

model that uses coe�cients from more powerful cross-sectional estimation, while Simin (2008) and

others �nd much less predictive power using less powerful time series estimation.

The paper proceeds as follows. Section 2 develops the investor's security selection problem, and

considers the relevance of the factor and score anomalies that emerge from standard asset pricing

tests for portfolio choice. Section 3 derives the asymptotic and small-sample power of score and

factor anomaly tests. Section 4 concludes.

2 The Relevance of Standard Asset Pricing Tests

There are three potential audiences for asset pricing tests. The �rst, rational asset pricing, considers

anomalies to be a misspeci�cation of the risks that are relevant to the representative investor. If a

characteristic reliably predicts stock returns, it must be compensation for risk. The factor returns

covary with some underlying state variable that drives investor utility. New anomalies, if they are

deemed to be robust, are added to the set of known risk factors. With the presumption that risk

covariances are at the root of all seeming anomalies, rational asset pricing has necessarily focused

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on time-series intercept tests. The second audience, behavioral �nance, considers anomalies to be

examples of mispricing, driven by some combination of less-than-fully-rational preferences and limits

to arbitrage. The third audience, practitioners in investment management, considers anomalies to

be potential sources of risk-adjusted return that can improve the welfare of their clients in partial

equilibrium.

While our focus is on relevance to the third audience, it is worth saying a few words qualitatively

about the second. A candidate anomaly that passes the cross-section test but not the time series test

is arguably of academic relevance. In particular, the limits to arbitrage and so-called intermediary

asset pricing stipulates that shocks to arbitrageur or intermediary capital can make the returns to

seemingly unrelated anomalies correlated in the time series. If the goal is to understand investor

preferences or beliefs, then a characteristic that is uniquely useful in explaining the cross-section of

returns, but is spanned by other stronger anomalies in the times series is nonetheless relevant for

behavioral �nance. A full exposition of this argument is beyond the scope of this paper.

We apply the classic portfolio choice model of Markowitz (1952) to the problem facing the third

audience - an investor who cares about single period portfolio returns and variances. Rather than

attempting to characterize the general equilibrium in the spirit of Tobin (1958) or Sharpe (1964) or

Lintner (1965) that arises if all investors were rational and had these preferences, we stay in partial

equilibrium. We are interested in the case where active portfolio management can deliver superior

investment decisions for our non-representative investor. This happens when our investor has a

di�erent view of risk and return from the representative investor, either because of di�erences in

preferences or beliefs. It is worth noting that mean-variance portfolio choice is commonly used by

practitioners. For example, the portfolio construction software developed by MSCI, Axioma, and

North�eld all use some form of myopic mean-variance optimization, with constraints and non-linear

transaction costs.

In this context, our de�nition of an anomaly is simple: It is a set of scores, for each security in

the opportunity set, that is relevant for our investor's portfolio choice. If our investor can safely

ignore a set of scores, there is no anomaly. If our investor chooses to use this set of scores in his

decision making, then there is an anomaly. We build intuition in three steps. The �rst is the classic

case where our investor is risk averse and faces no trading frictions. The second is where our investor

becomes risk neutral but faces a simple form of transaction costs that are constant across securities.

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And, the �nal step combines both risk aversion and transaction costs. These help establish the

applicability of two standard asset pricing tests: the cross section Fama and MacBeth (1973) test

popularized by the Fama and French (1992) assessment of anomalies; and the time series Jensen

(1968) alpha test popularized with the introduction of the Fama and French (1993) three factor

portfolio.

We also consider three extensions to the basic model in the Appendix A that allow: for trans-

action costs that vary across securities; for varying levels of assets under management; and, for

dynamic trading of the sort in Garleanu and Pedersen (2013). These extensions drive a wedge

between gross and net returns that varies across anomalies and thereby suggest straightforward

modi�cations to the two standard asset pricing tests that have the e�ect of netting out execution

costs. In some cases, these adjustments are dependent on the level of assets under management,

making the relevance of a particular anomaly context dependent, which is why we analyze them as

extensions.

2.1 The Return Generating Process: Scores and Factor Returns

We suppose that returns for N securities follow a linear factor structure at discrete times t ∈

{0, 1, . . . , T}.

rt = Γtft + εt

ft ∼ N (µ,Σ) εt∼N(0, σ2I

) (1)

The vector of individual security returns r, measured in excess of a risk-free rate of return, is

governed by a matrix Γ consisting of K < N row vectors of scores γ ′ that vary across securities i

at a time t, and a vector of normally distributed factor returns f that vary over time but not across

securities. The return of any security is equal to the sum product of its scores and corresponding

factor returns plus a residual idiosyncratic return. The K factor returns can be thought of as returns

to portfolios of stocks that can be estimated with scores and observed returns, ft = (Γt′Γt)

−1 Γt′rt.

What we refer to as scores are sometimes called characteristics in the academic literature. The

canonical Fama and French (2015) characteristics now include the ratio of book to market value,

the �rm's market capitalization, the annual rate of growth in assets, and operating pro�tability

scaled by assets, each transformed into buckets with common scores to limit the e�ect of extreme

6

scores. To this, many researchers add stock price momentum, typically measured as the most recent

annual return excluding the most recent month.

The assumption of normality in Equation 1 and a constant investment opportunity set, with

no time subscripts on µ or Σ aligns with the single-period mean-variance portfolio choice problem

that we evaluate in the next subsection. Unlike essentially all of our other assumptions about the

return generating process, these two come at the expense of generality. No doubt, the investment

opportunity set changes over time and some factor returns are not normally distributed, and partial

equilibrium investors likely care about timing factor returns and about the higher moments of their

portfolio returns.

Below, we will use examples with two factors at a time to build intuition, and will often leave

o� the subscript t to simplify the notation:

ri = γa,ifa + γb,ifb + εi (2)

We can map this into Equation 1:

r =

r1

r2...

rN

Γ =

γa,1 γb,1

γa,2 γb,2...

...

γa,N γb,N

f =

fa

fb

ε =

ε1

ε2...

εN

(3)

To simplify the analysis, we often de�ne the scores in a particular way, roughly in the spirit of

Fama and French, to make them analogous to portfolio strategies. The �rst column of scores γa is

equal to 1 for all �rms. The rest are de�ned so as to sum to zero. This means that the �rst factor

return fa will be the average or market return on all securities. Under the capital asset pricing

model (CAPM), for example, where returns are governed by a single factor, the second column of

scores γb is the demeaned CAPM beta β, equal to the standard CAPM beta less 1, and the second

factor return is equal to the �rst, fa = fb = rm, so that r = βrm + ε.

We are not otherwise specifying the distribution of the scores Γ, so in principle this setup could

accommodate sorted portfolios of the type in Fama and French (1993) or continuous variables of the

type in Fama and French (1992). There is a large literature on the relative merits of sorted portfolios

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versus more continuous or optimized weights. Hoberg and Welch (2009) argue that test portfolios

and factor portfolios are better constructed via optimization than via sorting. Daniel and Titman

(1997) and Davis, Fama, and French (2000) further consider whether sorting by characteristics and

covariances helps to resolve the age-old question of whether a �rm characteristic that is correlated

with future returns is a risk factor or mispricing. We side-step both of these issues, but our analysis

is closer in spirit to Hoberg and Welch (2009). In addition to possible improvements in explanatory

power that they document, we �nd linearly orthogonalized portfolios easier to work with analytically.

2.1.1 Exposures to Raw and Unit Factor Portfolios

A portfolio is de�ned by its vector of weights w on the available securities. It will be useful for

us to characterize any portfolio's exposure to two sets of factor portfolios, denoted by the matrices

Qraw and Qunit. We refer to the matrix Qraw = Γ as the set of raw factor portfolios that convert

the K sets of scores directly into portfolio weights. In general, as we have de�ned the scores above,

the �rst raw factor portfolio is the market portfolio, and the subsequent raw factor portfolios are

dollar neutral portfolios that tilt toward �rm characteristics, such as β. The weights in these raw

portfolios can in principle be correlated in the cross section. (The canonical Fama and French �rm

characteristics are correlated, to some extent, in their �nal formulation.) We will also refer to the set

of portfolios that are orthogonal, or cross-sectionally uncorrelated with all but one of the raw factor

portfolios, with the remaining covariance designed to be exactly one. This is the set of unit factor

portfolios that can be obtained by cross sectional regression of returns on scores, Qunit = Γ (Γ′Γ)−1.

Any portfolio w can be expressed as a linear combination of either raw or unit factor portfolios

plus an orthogonal residual η using a multivariate regression of the portfolio weights on the matrix

Q. This results in a vector e (w) of K multivariate exposures to the factor portfolios Q.

w = Qe + η ⇒ e (w) =(Q′Q

)−1Q′w (4)

The function eraw takes any set of portfolio weights w as an input and uses the matrix of

raw factor portfolios Q = Qraw, while the function eunit uses the matrix of unit portfolio weights

Q = Qunit.

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eraw (w) = (Γ′Γ)−1 Γ′w

eunit (w) = Γ′w(5)

Note that the full set of unit factor exposures in the matrix Eunit of the set of unit factor

portfolios Qunit is, as designed, equal to the identity matrix. Each unit factor portfolio has exactly

unit exposure to a single factor and zero to the rest. Meanwhile, the unit factor exposure of the raw

factor portfolios has o� diagonal elements that are not zero. Raw factor portfolios can in principle

have di�erent gross exposure and also have incidental correlations among each other.

Eunit (Qunit) = Γ′Qunit = I

Eunit (Qraw) = Γ′Qraw = Γ′Γ(6)

2.1.2 Computing Portfolio Expected Return and Variance

A portfolio's realized return can be characterized as the product of its unit factor exposures and the

realized factor returns plus a residual return. Its expected return is the product of its unit factor

exposures and the expected factor returns. Portfolio variance can be computed analogously.

E (r′w) = E (f ′Γ′w+ε′w) = E (f ′eunit (w) + ε′w) = µ′eunit (w)

var (r′w) = var (f ′Γ′w + ε′w) = var (f ′eunit (w) + ε′w) = eunit (w)′Σeunit (w) + σ2w′w

(7)

With the return generating process in Equation 1, both expected return and variance can be

computed parsimoniously with the knowledge of factor exposures and the distributional properties

of factor returns. This is because the residual variance will often be small for large and diversi�ed

portfolios as N becomes large, but the risk from the factor covariance matrix remains.

eunit (w)′Σeunit (w) + σ2w′ww′w→0−−−−−→ eunit (w)′Σeunit (w) (8)

It is important to note that the factor returns themselves are not necessarily uncorrelated, even

though they are returns to unit factor portfolios. They have unique exposure to a single set of

scores. But, it is quite possible, and often true in US data, that two unit factor portfolio returns

will be correlated with each other in the factor covariance matrix Σ.

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2.2 The Investor's Security Selection Problem

We consider a single-period investor, without limits on leverage or short-selling constraints, who

cares about mean and variance and knows the return generating process in Equation 1. In selecting

a portfolio, our investor faces a simple form of quadratic transaction costs, which act as limits on

position size:

maxw

E(r′w)− λ

2var(r′w)− θ

2w′w (9)

or, substituting the return generating process as simpli�ed in Equation 7:

maxwµ′eunit (w)− λ

2

(eunit (w)′Σeunit (w) + σ2w′w

)− θ

2w′w (10)

and, substituting the de�nition of the unit exposures of a given portfolio shown in Equation 5:

maxw

(Γµ)′w − 1

2w′(λΓΣΓ′ + I

(σ2 + θ

))w (11)

We acknowledge that all of these modeling assumptions come at the expense of generality. Most

investors care about more than just mean and variance, they face sundry portfolio constraints,

they have the ability to trade dynamically, and dynamic trade leads to more complicated e�ects of

transaction costs, changing scores Γt, and changes in the investment opportunity set µ and Σ. We

analyze some of these as extensions below.

We consider two special cases of this objective function. The �rst is where the aversion to risk

λ is equal to zero. In other words, the investor is risk neutral, but the non-zero transaction costs

that he faces cause his optimization problem to remain convex. The second is where transaction

costs θ are zero, but there is aversion to risk λ. This situation, where our investor can trade at no

cost, is the classic problem in the academic literature on mean-variance optimization. Practically

speaking, its outputs might apply approximately to an investor with low assets under management.

For investors with higher levels of assets under management, variable costs of trade limit position

sizes. Realistically, investors care about both execution costs and risk, but it is easier to build

intuition for the two separate cases before we consider the general case.

In extensions in Appendix A, we �rst replace θ with a vector of trading costs θ that vary across

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securities. We then consider limits on leverage and the resulting e�ects of assets under management

on investors facing the vector of cost parameters θ. And we �nally consider simple dynamics

in the spirit of Garleanu and Pedersen (2013) to capture the extra costs of factors whose value

decays over time. These three extensions highlight several intuitive notions of execution costs at

the level of factor portfolios, which can then be neatly characterized in closed form solutions and

examples. Rather than using the standard asset pricing tests applied to gross returns, the two asset

pricing tests must be applied to net of execution cost returns that depend on our investor's speci�c

circumstances.

2.3 Optimal Weights

The solution to the investor's problem in Equation 11 involves a tradeo� between risk, return, and

execution costs. The optimal weights on each security are a function of security scores Γ, expected

factor returns µ, the covariance of factor returns Σ, transaction costs θ, the investor's risk aversion

λ, and assets under management A. At optimal portfolio weights, the marginal bene�t of incremental

weight in each security is equal its marginal cost in the optimal portfolio:

Γµ =(λΓΣΓ′ + I

(λσ2 + θ

))w∗

⇒ w∗ =(λΓΣΓ′ + I

(λσ2 + θ

))−1Γµ (12)

We analyze two special cases, when transaction costs are zero and when risk aversion is zero,

and then proceed to the general case, over the next three subsections, before considering the case

of transaction costs that vary across securities in Appendix A.

2.3.1 Risk Neutral, Constant Transaction Costs

First, we consider the simplest case, where λ is equal to zero and risk considerations are unimportant.

Our investor is interested in maximizing returns net of transaction costs. Then, the optimal weights

from Equation 12 simplify to:

w∗tc =1

θΓµ (13)

Intuitively, the optimal weight for an individual security is high when it scores well on factors

that have high expected returns. To get more visibility into the optimal weights, we can compute

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the exposure e of this portfolio w∗tc to the raw and unit factor portfolios using Equation 5.

eraw (w∗tc) = 1θµ

eunit (w∗tc) = 1θΓ′Γµ

(14)

The upshot is that our risk neutral investor's problem can be reduced constructing the raw factor

portfolios Qraw = Γ, learning the magnitude of the expected factor returns µ, and using this vector

as weights on the K raw factor portfolios, which are the columns of Qraw. If the second factor were

to have a zero expected return µb = 0, it can be ignored in the optimization problem, regardless of

its risk properties. What is trivially absent is Σ. If a factor has a positive expected return µb > 0,

but it has a zero alpha with respect to an existing factor c, the standard logic of Fama and French

(1993) says this is not a distinct anomaly. If factor b has a zero expected return µb = 0, but it has

a non-zero alpha, hedging an existing factor c with positive expected return µc > 0, the standard

logic of Fama and French (1993) says this is a distinct anomaly. But, for an investor who cares

only about minimizing transaction costs, factors are anomalies if and only if they have a non-zero

expected return in the sense of µb.

So, what to make of the exposure to unit portfolios in the second half of Equation 14? The

unit exposures depend not only on expected factor returns but also on the correlation structure of

scores. These are the unintended common risks of the raw portfolio exposures in the �rst half of

Equation 14. These exposures to unit portfolios are interesting, but not relevant to the investor's

optimization problem. For a risk neutral investor, the optimal portfolio inherits unintended but

irrelevant risk exposures. Hedging these unintended risks requires transaction costs and is therefore

suboptimal.

Portfolio Choice With Transaction Costs: For a risk neutral investor, identifying

anomalies relies only on procedures like Fama and MacBeth (1973), using the results of

papers that are in the spirit of Fama and French (1992) to test the signi�cance of µ:

Cross Section Test: µ = f =1

T

∑t

ft =1

T

∑t

(Γ′tΓt

)−1Γ′trt = 0 (15)

If a given factor γa passes this test of µa 6= 0, we call it a score anomaly. Performing factor

regressions in the spirit of Fama and French (1993) on the resulting factor portfolios will lead to

mistakes in factor selection and the search for anomalies, excluding valuable anomalies and including

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apparent anomalies that are valuable only for their risk properties Σ and not their expected returns.

For the risk neutral investor facing transaction costs, the search for anomalies starts and ends with

score anomalies.

Example 1

Consider an example of two factors, call them CAPM beta and operating pro�tability. (We are

ignoring the market portfolio for the moment to keep K = 2.) Suppose that these two factors have

the following structure, so that operating pro�tability and beta have a negative correlation from a

common component c, with the residual components a and b uncorrelated with each other.

γa,i =OP

A i= ci + ai (16)

γb,i = βi − 1 = −ci + bi (17)

Suppose the expected returns of the two factor portfolios are roughly equal to their average

returns in US data, so that µa > 0 and µb = 0. The optimal stock weight is proportional to its own

mean return:

w∗tc,i =1

θ(γa,iµa + γb,iµb) =

1

θγa,iµa (18)

The optimal portfolio is a scaled version of the raw portfolio a that tilts towards �rms with

high operating pro�ts and away from �rms with low operating pro�ts. The absolute magnitude of

the weights depends on the expected factor return µa net of transaction costs θ. When the ratio of

return to cost is high, the weights are correspondingly large. When the ratio of returns to cost is

low, the weights are correspondingly small. This is in loose terms a re�ection of the capacity of the

strategy in light of execution costs.

No information about CAPM beta is needed to form the optimal portfolio. To this investor,

CAPM beta is dead, and can be safely ignored, because it does not pass a signi�cance test that

rejects µb = 0 in the sense of Fama and French (1992). But, the exposure of the optimal portfolio to

the unit CAPM beta portfolio b reveals that the optimal weights have an incidental exposure, which

here comes from the common component c. For the risk neutral investor, this incidental exposure

is irrelevant in setting weights. Our investor could neutralize the exposure to CAPM beta, but he

13

chooses not to. The intuitive rationale is that hedging this zero return exposure raises transaction

costs without increasing the investor's utility. The results of this choice can be seen in the raw and

unit exposures of the optimal portfolio, using Equation 5 and using notation var (a) = s2a:

eraw (w∗tc) = 1θµ =

1θµa

0

eunit (w∗tc) = 1

θΓ′Γµ = 1

θ

s2c + s2a −s2c

−s2c s2c + s2b

µa

0

=

1θ

(s2c + s2a

)µa

−1θs

2cµa

(19)

The second entry in the unit portfolio exposure vector shows the negative beta tilt that comes

incidentally from exploiting operating pro�tability. Meanwhile, the second entry in the raw portfolio

exposure vector shows that market beta is not a score anomaly in this two factor example for a risk

neutral investor.

2.3.2 Risk Averse, No Transaction Costs

Next, we consider the case where transaction costs θ are equal to zero but our investor is risk averse,

so that λ > 0. Then, the optimal weights from Equation 12 simplify to:

w∗ra =1

λ

(ΓΣΓ′ + σ2I

)−1Γµ (20)

This is the classic solution to mean-variance optimization, when the return generating process

is expressed with a linear factor structure. Weights are increasing in the individual stock expected

returns, which here can be expressed as the linear combination of factor scores and expected factor

returns. and weights are decreasing in the individual stock contributions to risk, which here can be

expressed as the product of factor scores and the covariance of factor returns ΓΣΓ′ plus idiosyncratic

risk σ2. To get more visibility into the optimal weights, we can compute the exposure e of this

portfolio w∗ra to the unit factor portfolio . This is easiest to do by rearranging the �rst order

condition in Equation 20 and substituting the de�nition of exposure to the unit portfolio from

Equation 5:

Γµ = λ(ΓΣΓ′w

∗ra + σ2w∗ra

)= λΓΣeunit (w∗ra) + λσ2w∗ra (21)

14

We can further rearrange, and take limits as the number of stocks N grows large. As this

happens the weight on any one security becomes small, and both sides of the equation go towards

zero, allowing us to derive a simple and intuitive expression for the exposure of the optimal portfolio

w∗ra , to the unit factor portfolio:

Γ (µ− λΣeunit (w∗ra)) = λσ2w∗ra → 0

⇒ eunit (w∗ra) = 1λΣ−1µ

(22)

The upshot is that our risk averse investor's problem reduces to a mean-variance optimization

of factor portfolios, when he can trade frictionlessly at θ = 0 with the number of available stocks

N large relative to the number of factors K. It no longer su�ces to learn µ. Now, the covariance

properties of the factor portfolios Σ are also relevant. If a factor has a positive expected return

µa > 0, but it has a zero alpha with respect to an existing factor b, the standard logic of Fama

and French (1993) says this is not a distinct anomaly, and indeed it is not for a risk-averse investor

who faces no transaction costs. If factor a has a zero expected return µa = 0, but it has a non-zero

alpha, hedging an existing factor b with positive expected return µb > 0, the standard logic of Fama

and French (1993) says this is a distinct anomaly. And, for a risk averse investor who does not care

about minimizing transaction costs, this is indeed a useful hedge.

Portfolio Choice With Risk Aversion: For a risk averse investor facing no trans-

action costs, the search for anomalies occurs in two steps. The �rst step is to use a

procedure like Fama and MacBeth (1973) to estimate both µ and Σ as:

µ = f = 1T

∑t ft = 1

T

∑t (Γ′tΓt)

−1 Γ′trt

Σ = var (ft) = 1T

∑t

(ft − f

) (ft − f

)′ (23)

The second step is to perform factor regressions in the spirit of Fama and French (1993)

on the resulting factor portfolio returns, including those with zero means µa = 0. This

will lead to the elimination of factor portfolios with positive means but zero alphas, and

lead to the resurrection of factor portfolios with zero means but non-zero alphas that

come from their useful hedging properties. Alpha here and above refers to a Jensen

(1968) alpha test. Sample averages are inserted into Equation 22, σ refers to elements

of Σ, and the null is that this �rst factor b is irrelevant in the choice of portfolio weights.

If under the null, eunit,b (w∗ra) = 0, this implies:

15

∑k 6=b

σbkeunit,k (w∗ra) = µb (24)

Equation 24 is equivalent to regressing the time series of fb on the time series of the

remaining factors excluding b, estimating the multivariate factor loadings β and testing

the signi�cance of the intercept:

Time Series Test: µb −∑k 6=b

βbkµk = 0 (25)

This derivation is in Appendix B. This means that some factors with µb = 0 can nonetheless

be factor anomalies because of their covariance properties mean that µb − βbaµa − βbcµc + · · · 6= 0.

Some factors with µb 6= 0 may nonetheless not be factor anomalies, because of their covariance

properties µb − βbaµa − βbcµc + · · · = 0. For the risk averse investor facing no transaction costs,

the search for anomalies starts and ends with factor anomalies. In our nomenclature, these remain

score anomalies, but they are not factor anomalies.

Example 2

Consider again two factors, as in Example 1, but the �rst is now the standard market portfolio, and

the second is still the CAPM beta, as before, with the mean return on the market factor µa > 0 and

the mean return on the beta factor µb = 0. Further, assume that the payo�s to the two factors are

positively correlated, so that the o�-diagonal elements of Σ are positive, so that σab > 0, meaning

that a portfolio that is long �rms with high betas has relatively higher returns when the market

also has relatively higher returns, as is true in US data. Plugging in to Equation 22:

eunit (w∗ra) =

eunit,rm

eunit,β

=1

λ

σ2bb

(σ2aaσ

2bb−(σab)

2)µa

− σab(σ2

aaσ2bb−(σab)

2)µa

(26)

The optimal strategy involves exposures to the market portfolio and the unit beta portfolios.

The absolute magnitude of the exposure to the market portfolio is increasing in its expected factor

return µa and decreasing in its expected favor risk σ2aa, so roughly speaking increasing in the Sharpe

ratio of the market, and this exposure is further increased because its risk can be mitigated with a

short position in the unit beta portfolio. Unlike the case of a risk neutral investor facing transaction

costs, this exposure is worth hedging despite its zero mean return, because it lowers risk and is

16

costless to execute. So, in that sense the early reports of the death of market beta are exaggerated.

It is very much alive as a factor anomaly, relevant for a risk averse investor facing no transaction

costs, in this two factor example.

Example 3

Consider again an example of two factors in Example 1, now call them the ratio of book to market

equity and low asset growth. Suppose that the mean returns are positive µa, µb > 0 as they are

in US data. Further, assume that the payo�s to the two factors are positively correlated, so that

the o�-diagonal elements of Σ are positive, so that σab > 0, meaning that a portfolio that is long

�rms with high ratios of book to market equity has relatively higher returns when a portfolio that is

long �rms with low asset growth also has relatively higher returns. Now we have the more general

version of the unit exposures from Equation 22:

eunit (w∗ra) =

eunit, BM

eunit,−∆AA

=1

λ

σ2bb

(σ2aaσ

2bb−(σab)

2)µa − σab

(σ2aaσ

2bb−(σab)

2)µb

σ2aa

(σ2aaσ

2bb−(σab)

2)µb − σab

(σ2aaσ

2bb−(σab)

2)µa

(27)

It turns out that empirically, the �rst entry is indistinguishable from zero, because the alpha of

the unit book to market equity portfolio when low asset growth is included as a reference portfolio

is approximately zero: µa − σabσ2bbµb ≈ 0. In that sense, we could substitute out information on book

to market equity:

eunit (w∗ra) =1

λ

0

1σ2bbµb

(28)

So, the ratio of book to market equity is not a factor anomaly, and not relevant for a risk averse

investor facing no transaction costs, in this two factor example. The only information that is needed

is the unit low asset growth portfolio and its Sharpe ratio. Interestingly though, the ratio of book

to market equity remains a score anomaly, relevant to an investor facing realistic transaction costs,

as we see next.

17

2.3.3 Risk Averse, Constant Transaction Costs

The more general case involves both risk aversion and transaction costs, bringing us back to Equation

12. Intuitively, when transaction costs are small, the conclusions of the simple risk aversion case

apply. Transaction costs act much like idiosyncratic risk, which is an issue when the investment

opportunity set is small, but not when the number of securities N is very large. However, there

are reasons to believe that the e�ect of θ might be meaningful even when the e�ect of idiosyncratic

σ2 is small. When assets under management are large, even a small weight in a given stock might

be large in comparison to its trading volume. Recall that Fama and MacBeth tests as in Equation

15 are necessary and su�cient for identifying anomalies in the presence of transaction costs alone.

And Jensen's alpha tests as in Equation 25 are necessary and su�cient for identifying anomalies in

the presence of risk aversion alone. When our investor is risk averse and faces transaction costs,

anomalies of both types are relevant. We can start with a variant of Equation 22 where we assume

that the considerations of idiosyncratic risk are second order but transaction costs remain �rst order,

so that θ ≫ λσ2:

Γ (µ− λΣeunit (w∗)) =(λσ2 + θ

)w∗ → θw∗

⇒ eunit (w∗) = 1λΣ−1 (µ− 2θeraw (w∗))

(29)

While it is obviously unappealing to have optimal weights and exposures on both sides of the

equation, Equation 29 is a useful intermediate relationship. Note that the optimal unit exposures are

driven by the standard risk and return considerations, but the return is haircut by the transaction

costs associated with the raw exposures of the optimal weights. The intuitive appeal is that we can

think of the risk averse investor optimizing over risk and net of transaction cost return.

We can eliminate the dependence using the de�nition of raw exposures from Equation 5:

eunit (w∗) =[λΣ + θ

(Γ′Γ

)−1]−1µ (30)

We examine the intuition in Equation 30 in the example below, but it is apparent that the

Jensen's alpha test will no longer be necessary for a factor to be a relevant anomaly. And, in

general, factors that pass either test will be worthy of consideration.

Portfolio Choice With Risk Aversion and Transaction Costs: For a risk averse

investor facing transaction costs, the search for anomalies means �nding factors that

18

pass either the test in the spirit of Fama and MacBeth (1973) in Equation 15 or the test

in the spirit of Jensen's alpha test in Equation 25.

Example 4

Consider the same setup as Example 3, with the ratio of book to market equity and low asset

growth. Recall we found that the ratio of book to market was a score anomaly, because it has a

mean return greater than zero µa > 0, but not a factor anomaly, at least controlling for low asset

growth, because its Jensen's alpha is zero, with µa − σabσ2bbµb ≈ 0. Further, assume that book to

market equity and low asset growth scores, γa and γb are constructed so as to be orthogonal to one

another, with unit standard deviation, which has the notation bene�t of making Γ′Γ = NI. Note

that the correlation of factor returns Σ can still be high, even if the scores are uncorrelated. Now,

we start from the result in Example 2, Equation 28 and generalize the unit exposures to include

transaction cost e�ects:

eunit (w∗) = 1λC

σ2aa + Nθλ σab

σab σ2bb + Nθλ

−1 µa

µb

= 1

λC

σ2bb

(µa − σab

σ2bbµb

)+ Nθ

λ µa

σ2aa

(µb − σab

σ2aaµa

)+ Nθ

λ µb

(31)

We substitute C ≡((σ2aa + Nθ

λ

) (σ2bb + Nθ

λ

)− (σab)

2)to save space. The optimal exposure to

the unit portfolios is a balance of two concerns. If the diagonal of factor risk Σ is high and risk

aversion is high, then the solution looks like Equation 22, while if the transaction costs θ are high,

then the solution looks like Equation 14. And, in general, the optimal solution is a blend of these

two concerns: managing portfolio risk-adjusted return while keeping execution costs low. Even

substituting in the implications of a Jensen's alpha of zero for the book to market factor portfolio,

there is still positive portfolio exposure to this factor, because of its raw factor return µa:

eunit (w∗) =1

λC

Nθλ µa(

σ2aaσ

2bb−(σab)

2

σ2bb

+ Nθλ

)µb

(32)

So, the ratio of book to market equity is not an anomaly for a risk averse investor facing no

19

transaction costs in this two factor example, but it is resurrected in the case involving transaction

costs. Our investor tilts somewhat toward the lower alpha unit portfolio to access raw returns at

lower cost. The general conclusion is that the investor will put weight on both factor and score

anomalies.

2.4 Summary

Suppose a candidate factor appears: A �rm characteristic that has the potential to predict its stock

returns. Is it a relevant anomaly? The literature contains two standard asset pricing tests. The �rst

uses Fama and MacBeth regressions of the sort in Equation 15, testing whether the mean of the new

factor returns is equal to zero using a series of multivariate cross-sectional regressions that contain

other factors known to predict the cross section of stock returns. We show that this test is always

applicable for an investor facing a simple form of constant transaction costs, whether he is risk

averse or not. The second uses Jensen's alpha tests of the sort in Equation 25. These test whether

the intercept in a regression of the new factor portfolio returns, from the aforementioned cross-

sectional regressions, on the portfolio returns of existing factors. We show that this test is always

applicable for a risk averse investor, whether he faces transaction costs or not. As it turns out, both

are applicable for the general case of a risk-averse investor facing a simple form of transaction costs.

Realistic transaction costs complicate these conclusions, but in an intuitive way. When trans-

action costs vary across stocks, when assets under management are substantial, and when dynamic

trading considerations appear, each of tests should in principle be performed on factor returns that

are net of execution costs, and gross of future persistence in returns. Tests of statistical signi�cance

in the cross section test are still valid in a number of special cases. Appendix A provides some

illustrations of how this might be done in practice.

We now turn to the power of the two tests.

3 The Power of Standard Asset Pricing Tests

The previous section shows the relevance of both score and factor anomalies for portfolio choice.

In this section, we turn to the econometrics of identifying anomalies. There are two standard tests

in empirical asset pricing. Tests in the spirit of Fama and MacBeth (1973) and Fama and French

20

(1992) are cross-sectional and do not consider the covariance properties of factor portfolio returns.

Tests in the spirit of Jensen (1968) and Fama and French (1993) use time-series data and focus on

the covariance properties of the factor portfolios as well their means.

We take two approaches to estimating the power of these two tests. In the �rst, we compute

the asymptotic power curve analytically. The score test is more powerful by a multiplier that is

increasing in the Sharpe ratio of the factor portfolios. In the second, we simulate the data generating

process in small samples, and compute a simulated power curve. The e�ect of the Sharpe ratio is

magni�ed by a small sample estimation of the factor portfolio return covariances.

3.1 Asymptotic Power Curves

Section 2 considers the problem facing an investor who understands the return generating process

in Equation 1. Realistically, the investor does not know the expected payo�s to factor portfolios µ

or their covariances Σ. He must use data on scores and stock returns to estimate these. And, in

practice, our investor also needs to consider whether market forces might make estimates irrelevant

for future returns. For now, we assume that our investor di�ers from the representative investor

in either preferences or beliefs, so that the history of stock returns can be used to produce reliable

forecasts of the parameters in Equation 1.

3.1.1 Data Mining

This raises the issue of data mining, which can take two forms: selection and over�tting. Problems

of selection stem from starting with n candidate anomalies and choosing k < n that work in the

historical data. Problems of over�tting come from optimally weighting n candidate anomalies using

their in-sample performance into one aggregate blended superscore. In both cases, the likelihood

of Type I error, where a �rm characteristic is deemed spuriously to be an anomaly, is high. These

issues are discussed in Mclean and Ponti� (2016), Harvey, Liu, and Zhu (2015), and Bailey and

López de Prado (2014). For example, Mclean and Ponti� (2016) examine the e�cacy of anomalies

using realistic rolling estimations of factor average returns and the incremental e�ect of publicizing

the �nding. Simin (2008) and Levi and Welch (2014) �nd that rolling estimates of expected returns

from the Fama-French three factor model do not work well at forecasting return realizations. Their

�ndings point to researcher data mining. Novy-marx (2016) computes adjusted t-statistics as a

21

function of n, k, and the nature of the over�tting procedure employed. Given the number of

researchers focused on asset pricing and the academic and commercial incentives for documenting

anomalies, the issue of Type I errors is paramount.

However, Lewellen (2015) �nds that rolling estimates using characteristics as in our return

generating process in Equation 1 produce a t-statistic greater than 10. Moreover, it is worth

considering the power to detect anomalies in the �rst place, and the possibility of Type II errors,

which is the focus of this paper. For example, it is possible to turn the argument in Novy-marx

(2016) around. Suppose the objective was not to establish an anomaly but rather to overturn an

existing anomaly. A similar logic then applies. By starting with n potential controls or established

anomalies and risk factors and choosing k < n in the historical data, it is possible to lower the

power a test of the risk anomaly. This is particularly true in time series tests, as we argue below,

and Novy-marx (2012), who argues that the risk anomaly is subsumed by pro�tability and value,

is an example of this approach.

In this paper, we sidestep the issue of data mining and imagine that there is an established set

of anomalies to which a single new candidate may be added. We start with cross sectional tests,

which are relevant for an investor facing material transaction costs.

3.1.2 The Cross Section Test

The investor, or the econometrician, estimates Equation 1, where we assume that the �rst factor is

the market portfolio, so that γa = 1, with a series of cross sectional regressions:

rt ∼ Γtft + εt ⇒ ft =(Γ′tΓt

)−1Γ′trt (33)

The time series mean of the factor payo�s is the Fama and MacBeth estimator of the mean

payo� µ:

µ =1

T

∑t

ft (34)

In a Bayesian sense, all factors will be relevant in at least a small way, in the sense that the point

estimate for the return µb on a factor b will never be exactly zero. But, in a frequentist sense, there

are some factors that cannot be statistically deemed relevant. This is the usual notion of a score

22

anomaly: We must be able to say that the hypothesis that µb 6= 0 is true with su�cient probability.

The standard error of the estimate of µb is equal to:

se (µb) =1√T

√σ2bb +

1

N2σ2∑i

(δ′bγi

)2(35)

Where σ2bb is the variance of factor b from the factor covariance matrix Σ, N is the number of

�rms in the cross section, and σ2 is the idiosyncratic variance that is assumed for simplicity to be

constant across �rms. And, we de�ne the vector δb from the inverse of the matrix Γ′Γ as follows:

N(Γ′Γ

)−1=

[δa δb δc · · ·

](36)

We derive the expression for standard error in Equation 35 in Appendix B. Because we estimate

the time series mean of the regression coe�cients, we lose 1 degree of freedom and the estimate of

the asymptotic variance is

se (µb) =1√T − 1

√σ2bb +

1

N2σ2∑i

(δ′bγi

)2(37)

For a large cross section of stocks N, the e�ect of residual risk ε becomes small in the second

part of the expression under the radical. The estimate of the factor payo� in each period using only

returns rt and scores Γt at any given time t becomes very precise. The error in the estimate of the

mean then depends only on the number of time periods T. This is the asymptotic standard error

in the sense that σ2bb and σ2 are not known to the investor or the econometrician in small samples.

We consider the small sample properties with Monte Carlo simulations below. For now, we plot the

power curve using Equation 35 in Figure 1, using

N = [50; 100; 250; 1, 000; 10, 000]

T = [50; 100; 200; 500; 1, 000]

σε = [0; 3.27; 5; 6; 7]

σbb = [0.5; 1; 2; 3; 4]

(38)

Underlined values are the base case and represents by the blue line in the graph. Monthly

idiosyncratic risk σε and factor risk σbb are in percentage. We vary monthly factor return µb so that

23

its annual Sharpe ratio ranges from 0 to 2. Annual Sharpe ratio is calculated as SR = µbσbb·√

12.

The power curves illustrate the probability that the null of zero is rejected given a variety of

inputs for the true mean of factor return µb. It shows the power our investor has to detect relevant

score anomalies. The y-intercept is the size of the test. There is a 5% chance of rejecting the

null of a zero factor return when it is truly zero. In these situations, the investor concludes that

there is a score anomaly when in truth there is not one. The shape of the power curve is otherwise

not terribly interesting, as it depends on the assumed distribution of factor scores, the number

of separate periods T, the assumed factor variance σ2bb, and the assumed number of �rms N, and

idiosyncratic risk σ2. These shift the power curve in intuitive ways. The power rises more steeply

with more �rms in Panel (a) and lower idiosyncratic variance in Panel (b). A larger number of �rms

in the cross-section helps to eliminate the e�ect of idiosyncratic risk on factor returns. A smaller

amount of idiosyncratic risk has a similar e�ect, in that even a small number of �rms deliver a

pure factor return. In both cases though, the improvements in power are limited. Even an in�nite

number of �rms does not lead to an extremely powerful test, capable of detecting small anomalies.

The power also rises more steeply with more time periods in Panel (c) and lower factor variance

in Panel (d). A larger number of time periods means that the mean factor return per period can

be estimated with greater and greater accuracy, assuming there are no changes in the underlying

return generating process. Similarly, power rises more quickly if the factor payo�s are very reliable,

falling very close to the mean in every period. These are situations where a score anomaly can

be reliably detected even when the true average return is quite small. All four panels show that

economically large score anomalies are always detected even in modest time series, but these will

be rare in competitive markets, so power is important.

We are more concerned with the relative power of the cross section and time series tests, which

we turn to next, than we are about the other comparative statics, which will improve the power of

both proportionally.

3.1.3 The Time Series Test

We next move to the time series test, which is relevant to an investor who is risk averse. In this

case, there is a second step after the estimation of factor payo�s in Equation 33. Practically, our

investor is interested in whether a particular factor will have zero e�ect on his portfolio choice in

24

Equation 22. And, as we argue above, this is equivalent to the factor passing a Jensen (1968) alpha

test in Equation 25. We leave o� the hats on the factor returns in the regression of the returns of

a given factor b on the remaining factors other than b:

fb,t ∼ αb + β′bf−b,t + εt ⇒ αb = fb − β′bf−b = µb −′ β

′bµ−b (39)

We use the subscript to indicate the full set of factor returns f−b or means of the factor returns

µ−b to indicate the vectors that exclude the factor b. The factor loadings are the vector β−b, and the

variable of interest is the factor-risk-adjusted return αb. This is an intercept test to see whether the

factor return µb is large enough given its covariances with the other factors in the set of anomalies.

Again, in a Bayesian sense, all factors will be relevant in some small way. In a frequentist sense,

we are interested in ruling out factors that are statistically irrelevant. This is the usual sense of a

factor anomaly, that we must be able to say that the hypothesis that αb 6= 0 is true with su�cient

probability. The standard error of the estimate of αb is equal to:

se (αb) = 1√T

√σ2ε(1 + µ′−bΣ

−1−bµ−b

)= 1√

T

√(σ2bb + 1

N2σ2∑

i

(δ′bγi

)2)(1−R2) (1 + SR2)

(40)

We derive the expression for standard error in Equation 40 in Appendix B. Because in the time

series regression we estimate theK+1 coe�cients, we loseK+1 degrees of freedom and the estimate

of asymptotic variance is

se (αb) = 1√T−K−1

√σ2ε(1 + µ′−bΣ

−1−bµ−b

)= 1√

T−K−1

√(σ2bb + 1

N2σ2∑

i

(δ′bγi

)2)(1−R2) (1 + SR2)

(41)

For a large cross section of stocks N, the e�ect of residual risk ε becomes small and the variance

of the residual factor return risk ε is simply σ2bb. The estimate of the factor payo� in each period

using only returns rt and scores Γt at any given time t becomes very precise. When the cross-section

of �rms N is not so large, then the variance is greater, so that σ2ε > σ2bb, and equals the quantity

under the radical in Equation 35, discussed above. In addition to these comparative statics, there

are two additional drivers of the standard error of the time series test. The standard error now

depends on the means and covariances of the other factor returns too. The �rst term in parentheses

25

is an increase in power that comes from the fact that the other factors can in principle reduce the

residual variation in the regression equation. The residual in the time series test is smaller than the

residual in the cross section test by an amount equal to one minus the time series R-squared. The

second term in parentheses is decrease in power, equal to one plus the in-sample maximum squared

Sharpe Ratio (SR) of the other factor returns. This is the mean-variance optimal combination of

the existing factors. When the existing factors are very powerful predictors of return, then the

standard error in Equation 40 rises, as in Shanken (1992). It is harder to reject the null. This

is the asymptotic standard error again in the sense that σ2bb, σ2, µ−b, and Σ−b are not known to

the investor or the econometrician in small samples. We consider the small sample properties with

Monte Carlo simulations below. For now, we plot the power curve using Equation 40 in Figure 2,

using the same parameters as in Figure 2 with one more parameter to vary:

SR = [0.25, 0.42, 0.50, 0.75, 1] (42)

Underlined values are the base case and represents by the blue line in the graph.

The power curve exactly mirrors the results in Figure 1, but with the power shifted down by the

in-sample Sharpe ratio. As before power rises more steeply with more �rms and lower idiosyncratic

variance. The gains from these two parameters are bounded. Power rises more quickly with more

periods and lower factor variance. With a large number of periods and low factor variance, the time

series test can detect small anomalies.

To these comparative statics, we now add the Sharpe ratio of the existing factors in the next

section. While it is not immediately apparent in the comparison of Figure 1 and Figure 2, the time

series tests are all shifted down somewhat.

3.1.4 A Comparison of the Cross Section and Time Series Tests

It is immediately apparent that the standard error in Equation 40 contains a potential loss in power,

when compared to the standard error in 35. If a new factor has no true connection to the time

series payo�s of the incumbent set, so that the R-squared in 40 is zero, then the standard errors are

related by the following formula:

26

se (αb) = se (µb)√(

1 + µ′−bΣ−1−bµ−b

)(43)

Correspondingly, the relation between estimates of asymptotic standard error is

se (αb) = se (µb)

√T − 1

T −K − 1

√(1 + µ′−bΣ

−1−bµ−b

)(44)

In other words, it is harder to �nd a new factor anomaly than it is to �nd a new score anomaly.

We overlay the power curves in Figure 1 onto Figure 3 at various Sharpe ratios.

This has a nice intuition. When the predictive power of existing factors is large relative to the

portfolio variance - the Sharpe ratio is high - the estimation of a covariances β−b between the new

factor b and existing factors becomes a source of error that drives power down relative to the cross-

section test. It is possible to connect the new factor to existing ones in a way that is spurious. For

example, both momentum and CAPM beta su�ered very poor returns in the market reversal of the

spring of 2009, but are otherwise essentially uncorrelated. Similarly much of the overlap between

CAPM beta and the ratio of price to book occurs in the late 1990s and early 2000s. If we consider

the Sharpe ratio of momentum and the price-to-book to be high - as they are in US data - and for

these to be a spurious correlations - which may be true - then it is easier to reject CAPM beta as

a factor anomaly. It is this possibility which lowers the relative power of the test. In this sense,

our argument is related to Ang, Liu, and Schwarz (2008). They focus on the loss of power that can

come from aggregation. Forming portfolios improves the estimation of covariances but throws away

information. In our context, it is the extra need - for risk averse investors - to compute covariances

that diminishes their ability to �nd relevant anomalies.

3.1.5 The Sharpe Ratio in US Data

This begs the question: Which line in Figure 3 in the comparison of cross section and time series

tests is the relevant one? This depends on the size of the in-sample Sharpe ratio for some set of

existing factors, like the Fama-French �ve factors. Power falls monotonically in two ways: as the

factor set increases, and as the horizon rises, from daily to weekly to monthly to annual returns.

First, a larger factor set by de�nition means a higher in-sample Sharpe ratio. The Fama and

French (2015) �ve-factor model, for example, will mechanically reject more potential factor anoma-

27

lies than the Fama and French (1993) three-factor model. So, there is a practical limit on the

number of factor anomalies that can be discovered. By contrast, the number of score anomalies

that can be discovered is only limited by the number of stocks N and the number of periods T,

but it is not otherwise constrained by the predictive power of existing cross-sectionally orthogonal

anomalies.

Second, a property of the Sharpe ratio is that it rises as the return horizon increases. This is

because it is the ratio of mean to standard deviation. While the mean increases linearly in T , the

standard deviation increases linearly in√T .

The qualitative impact of these two e�ects is clear. We use US data to convert qualitative

to quantitative e�ects on econometric power. The Sharpe ratio of the set of standard, existing

factors is reasonably high, when we use monthly returns, as is common practice in the academic

literature. The bold line in Figure 3 shows the loss in power of a test that uses a standard set of

the Fama-French �ve factors, momentum, and short-term reversal and a monthly return horizon.

The empirical moments of these portfolios over the period from 1963:07 to 2016:03 are shown in

Table 1. In Table 1, we also include a Fama-French style portfolio using market beta, using the beta

estimation approach of Frazzini and Pedersen (2014). This portfolio divides the CRSP universe into

small and large, using the median size among NYSE stocks as the breakpoints and further divides

small and large stocks into three terciles according to market beta, again using NYSE breakpoints.

The individual annual Sharpe ratios range from 0.29 to 0.56, and the in-sample optimal annual

Sharpe ratio of the seven factor returns, excluding market beta, is 1.45. The corresponding quarterly

Sharpe is 0.73, the corresponding monthly Sharpe is 0.42 which is what we show in bold in Figure

3, and the corresponding daily Sharpe is 0.09.

3.1.6 Choosing Return Horizon

Plugging these Sharpe ratios into Equation 43, we can compute the loss in power. To �x ideas, we

use T = 200, N = 250, σbb = 3, σε = 3.27. For monthly tests, which is the standard in the literature

and which we indicate in bold in Figure 3, the maximum loss in power is 22.4 percent for a new factor

that has a true annual Sharpe ratio of 0.67 � in other words, a very strong anomaly. Annual return

horizons are rarely used, likely because they involve an intuitively dramatic reduction in power,

with the maximum loss at 73.0 percent. Quarterly returns are occasionally used. For example, see

28

Cederburg and O'Doherty (2016). Quarterly returns lie in between, with a substantial maximum

loss at 45.1 percent. This is an e�ective way to stack the deck against rejecting the null. Meanwhile,

daily returns largely solve the power problem with a loss in power of only 1.4 percent at maximum.

To provide an illustration, we compute alphas in Table 2 for six of the Fama-French style factor

portfolios summarized in Table 1. We leave out reversal, which has a payo� that is very short-lived.

We use monthly, weekly, and daily returns and the Fama-French �ve-factor model in Panels A, B,

and C, respectively. Our prior from Equation 43 and the properties of the Sharpe ratio as horizon

falls is that the daily tests will be the most powerful, delivering lower standard errors and higher

t-statistics on average. Table 2 bears this out. We are focused on the �rst two columns, which

display coe�cient estimates, standard errors, and t-statistics for the intercept, or alpha of each

portfolio. In the monthly tests in Panel A, the average standard error of the alpha estimate for

these six portfolios, at 1.25 percent annualized, is 25 percent higher than in the daily tests, shown

in Panel C. The average t-statistic, at 4.38, is 44 percent higher. Notably, the conclusion from Fama

and French (2015) that the price-to-book ratio is subsumed by the other factors is reversed in daily

data. HML has a t-statistic of 0 in monthly data and a t-statistic of 2.19 in daily data. It retains a

statistically signi�cant alpha of 1.96 percent annualized.

To illustrate the di�erence in power further, we repeat this exercise for the �ve-by-�ve portfolios

from Ken French's data library that double sort on size and book-to-market, pro�tability, and

investment. We focus on these three, because daily returns are available, and the sorting variables

are updated monthly, making comparisons among the horizons valid. We focus on the top and

bottom sets of �ve portfolios, where we expect to �nd alphas di�erent from zero, or 30 in all. For

each of these 30 portfolios, we conduct a time-series test using the Fama-French �ve-factor model

as in Table 2, excluding the factor of interest, recording the absolute annualized alpha coe�cient,

the annualized standard error, the absolute t-statistic, and the p-value for the resulting alpha, using

�rst daily returns, then weekly, and then monthly returns. We plot the cumulative distribution of

these four values in the four panels of Figure 4. The absolute annualized coe�cient ranges from 0.01

percent to 22.8 percent, the annualized standard error of the coe�cient ranges from 0.09 percent

to 0.51 percent, the absolute t-statistic ranges from zero to 25.3, and the p-value ranges from zero

to 1.0. Daily alphas reject the null of zero alpha relative to the Fama-French 5-factor model 63.3

percent of the time, relative to the 5% size of the test. Meanwhile, weekly alphas reject 56.7 percent.

29

And, monthly alphas reject only 40.0 percent. There are far more novel asset pricing anomalies to

be discovered in daily data.

Why not then simply use the more powerful daily, or better yet intraday, returns? There are

two reasons why the literature has tended to use monthly returns, beyond the ease of computation.

One is that higher frequency covariances may understate the true, tradeable covariances, because

of asynchronous correlation. For example, if some stocks simply do not trade every day, or trade

in a way that is slow to incorporate aggregate information, then the estimated betas to the Fama-

French �ve factors will be biased downward in daily regressions. [MB FILL IN Citations] Rather

than use monthly returns, Scholes and Williams (1977) recommend aggregating lagged and leading

covariances, e�ectively moving toward weekly regressions. The other is that daily returns may

understate or overstate the tradeable annual returns of a given anomaly, as emphasized by Liu

and Strong (2008). This is a closely related point. Instead of asynchronous correlation between

a given anomaly and the Fama-French portfolios, this is the autocorrelation of anomaly returns.

Positive autocorrelation, or anomaly momentum, means that annualized monthly returns are higher

than annualized daily returns. Negative anomaly autocorrelation, or anomaly reversal, means that

annualized monthly returns are lower than annualized daily returns. Both of these e�ects can cause

biased inference in identifying tradeable anomalies, moving from monthly to more powerful daily

regressions. Both of these are reasons that the analysis of shorter horizons might bias inference, but

it is important to note that shifting to much longer horizons comes at the expense of power.

Fortunately, it is possible to split these e�ects apart and make a modest suggestion for best

practice � at least if the anomalies portfolio from Ken French's data library that we analyzed in

Figure 4 are a useful guide. The p-value results in panel D of Figure 4 come from two separate

e�ects. The p-value is higher because of higher power � that is the reduction in average annual-

ized standard errors in Panel B. In addition, the p-value is higher because of potential bias from

asynchronous correlations and factor momentum and reversal � that is re�ected in the increase the

average annualized absolute coe�cient in Panel A. What is the right tradeo� between power and

bias? To answer this question, we plot the coe�cients and standard errors in Figure 5. We scale

the annualized coe�cient estimate for each horizon by the 50-day annualized coe�cient estimate,

on the argument that the bias arising from trading e�ects is negligible at that point. We scale the

annualized standard error by the 1-day estimate, where power is maximized. Panel A of Figure 5

30

shows the averages of these scaled values across our 30 portfolios as a function of return horizon,

from daily returns to overlapping 50-trading-day returns. We repeat the exercise using medians

instead of averages in Panel B.

Note that the average (and median) coe�cient decreases as we move from daily to ten days

and then levels o�. This in principle might re�ect biased inference, though it could also come

from a more accurate estimation of covariances. Meanwhile, the average (and median) standard

error continues to rise from daily horizons to 50-day horizons and beyond. If the data from Ken

French's data library are suggestive of typical anomalies, our analysis suggests that the problem of

bias for these factors appears to be largely solved at 10 trading days, or two weeks, and completely

solved by 20 days. At ten days, the maximum loss in power is 11.6 percent, which still represents a

considerable loss of power in time series tests relative to cross-sectional tests, but it is more modest

than in the standard practice of analyzing monthly returns, with a loss of 22.4 percent. In what

follows, we continue to use monthly returns, as the standard in the literature, but an important

conclusion is that, by shifting from monthly to two-week returns, there appears to be a free increase

in the power of standard asset pricing tests.

3.2 Small Sample Power Curves

The power curves in Figure 3 assume that all of the distributional parameters in the return gener-

ating process in Equation 1 are known by the investor. In practice, they are not. Sample estimates

must replace their respective population values. We estimate the small sample properties of Equa-

tions 35 and 40 by running Monte Carlo simulations.

In the base case, we consider N = 250 securities and T = 50 periods. We use the seven factor

portfolios discussed in last section. We �rst draw factor coe�cients Γi from normal distribution

N (0, 1) for each stock i. Then we draw factor returns from a multivariate normal distribution

N(µ, Σ

)with mean and variance estimated from US market data. Finally, we draw the idiosyn-

cratic return from a normal distribution N(

0, σ2ε

)with variance estimated from US market data

σε = 3.27% and calculate return of each stock. We repeat this Monte-Carlo simulation for 2000

times. We plot the simulated power curves against their theoretical values in Figure 6 and con�rm

they are consistent with each other. Then we plot the theoretical small sample distribution in

Figures 7, 8 and 9.

31

The patterns we see in the asymptotic power curves also hold in the small sample power curves.

Power of both tests get stronger as there are more securities, longer time periods, smaller variances

of factor and idiosyncratic returns. The number of time periods and the test factor's variance have

a more signi�cant e�ect on the power of both tests. The Sharpe ratio drives the wedge between

power of the cross section and time series tests. The di�erence in power for small sample is 23.7%

at SR = 0.69, which is larger than that for asymptotic result, 22.4%, because of additional loss of

degrees of freedom in the time series test.

4 Conclusion

What is an anomaly? Empirical asset pricing papers that aim to establish an anomaly often rely

on two types of tests. One is the cross-sectional test in the spirit of Fama and MacBeth (1973)

and famously used in Fama and French (1992), and the other is the times series test in the spirit

of Jensen (1968) and popularized by Fama and French (1993). We consider these tests from two

di�erent points of view: relevance and power. The cross-section test is relevant to a risk neutral

investor facing a simple form of transaction costs. The time series test is relevant to a risk-averse

investor facing no transaction costs. Meanwhile, both are relevant in the more general case of

risk aversion and transaction costs. Next, we show that the time series test can be inherently

lower powered. Given that most professional investors face meaningful transaction costs and most

commercial portfolio optimizers target a mean-variance objective, we believe that a test that passes

either of the two tests can be considered an anomaly - in the sense that it is practically relevant

for a large class of investors. Viewed in this light, the literature on empirical asset pricing has the

potential to identify a richer set of interesting anomalies than are contained in the Fama-French

3-Factor model or even the newer 5-factor model. Unlike the time series analysis in Fama and

French (1993), the cross-sectional framework in Fama and French (1992) has a higher upper limit

on the number of relevant and statistically signi�cant factors.

32

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35

Figure 1: Asymptotic Power Curve of The Cross Section Test

Note: The annual Sharpe ratio of the test factor is calculated as SR = µbσbb·√

12.

36

Figure 2: Asymptotic Power Curve of The Time Series Test

Note: The annual Sharpe ratio of the test factor is calculated as SR = µbσbb·√

12.

37

Figure 3: Asymptotic Power Curve of the CS and TS Test at Various Sharpe Ratios of the IncumbentFactor Model

Note: The annual Sharpe ratio of the test factor is calculated as SR = µbσbb·√

12.

38

Figure 4: Monthly, Weekly, and Daily TS Test Statistics

39

Figure 5: Bias and Power: Annualized Alphas By Return Horizon

40

Figure 6: Small Sample and Simulated Power Curve of the CS and TS Test

Note: The annual Sharpe ratio of the test factor is calculated as SR = µbσbb·√

12.

41

Figure 7: Small Sample Power Curve of the CS Test

Note: The annual Sharpe ratio of the test factor is calculated as SR = µbσbb·√

12.

42

Figure 8: Small Sample Power Curve of the TS Test

Note: The annual Sharpe ratio of the test factor is calculated as SR = µbσbb·√

12.

43

Figure 9: Small Sample Power Curve of the CS and TS Test at Various Sharpe Ratios of theIncumbent Factor Model

Note: The annual Sharpe ratio of the test factor is calculated as SR = µbσbb·√

12.

44

Table 1: Empirical Moments for Factor Returns (1963:07 to 2016:03)

(a) Mean and variance

MKT SMB HML RMW CMA MOM STREV RISK

Average 0.50 0.25 0.34 0.25 0.31 0.69 0.48 -0.22Annualized 5.98 3.05 4.11 3.06 3.71 8.29 5.81 -2.63

SD 4.45 3.06 2.85 2.12 2.01 4.24 3.13 4.65Annualized 15.41 10.62 9.89 7.35 6.96 14.67 10.84 16.10

Sharpe Ratio 0.11 0.08 0.12 0.12 0.15 0.16 0.15 -0.05Annualized 0.39 0.29 0.42 0.42 0.53 0.56 0.54 -0.16

(b) Covariance

MKT SMB HML RMW CMA MOM STREV RISK

MKT 19.75 3.82 -3.77 -1.93 -3.46 -2.44 3.96 14.79SMB 3.82 9.38 -1.01 -2.35 -0.70 -0.30 1.51 3.60HML -3.77 -1.01 8.13 0.53 4.02 -2.00 -0.10 -4.68RMW -1.93 -2.35 0.53 4.49 -0.36 0.82 -0.41 -1.97CMA -3.46 -0.70 4.02 -0.36 4.03 -0.08 -0.81 -4.19MOM -2.44 -0.30 -2.00 0.82 -0.08 17.92 -3.87 -4.16STREV 3.96 1.51 -0.10 -0.41 -0.81 -3.87 9.77 3.22RISK 14.79 3.60 -4.68 -1.97 -4.19 -4.16 3.22 21.57

45

Table 2: Time Series Tests: Monthly, Weekly, Daily

(a) Monthly

Intercept Mkt_RF SMB HML RMW CMA

coef se coef se coef se coef se coef se coef set t t t t t

SMB 3.65 (1.36) 0.13 (0.03) 0.06 (0.05) -0.42 (0.05) -0.12 (0.08)[2.68] [4.55] [1.06] [-8.23] [-1.52]

HML 0.00 (0.99) 0.02 (0.02) 0.03 (0.03) 0.14 (0.04) 1.01 (0.04)[0.00] [0.99] [1.06] [3.55] [23.44]

RMW 4.72 (0.98) -0.10 (0.02) -0.22 (0.03) 0.14 (0.04) -0.29 (0.06)[4.82] [-5.13] [-8.23] [3.55] [-5.10]

CMA 2.74 (0.66) -0.11 (0.01) -0.03 (0.02) 0.46 (0.02) -0.13 (0.03)[4.17] [-8.33] [-1.52] [23.44] [-5.10]

MOM 8.69 (1.98) -0.13 (0.04) 0.07 (0.06) -0.52 (0.08) 0.24 (0.08) 0.39 (0.12)[4.38] [-3.19] [1.16] [-6.64] [3.02] [3.29]

RISK -3.40 (1.54) 0.63 (0.03) 0.04 (0.04) 0.05 (0.06) -0.21 (0.06) -0.53 (0.09)[-2.20] [19.75] [0.86] [0.77] [-3.51] [-5.82]

Average 3.87 (1.25) 0.19 (0.03) 0.06 (0.03) 0.20 (0.04) 0.19 (0.04) 0.39 (0.06)[3.04] [6.99] [2.14] [5.91] [3.90] [6.53]

(b) Weekly

Intercept Mkt_RF SMB HML RMW CMA

coef se coef se coef se coef se coef se coef set t t t t t

SMB 3.63 (1.13) 0.00 (0.01) 0.01 (0.02) -0.41 (0.02) -0.07 (0.03)[3.20] [-0.25] [0.28] [-17.24] [-2.00]

HML 1.54 (0.97) 0.00 (0.01) 0.00 (0.02) -0.08 (0.02) 0.84 (0.02)[1.58] [0.31] [0.28] [-3.52] [36.07]

RMW 4.61 (0.84) -0.09 (0.01) -0.23 (0.01) -0.06 (0.02) -0.06 (0.02)[5.47] [-10.79] [-17.24] [-3.52] [-2.37]

CMA 2.80 (0.65) -0.12 (0.01) -0.02 (0.01) 0.38 (0.01) -0.03 (0.01)[4.30] [-20.04] [-2.00] [36.07] [-2.37]

MOM 8.23 (1.69) -0.07 (0.02) 0.05 (0.03) -0.64 (0.03) 0.13 (0.04) 0.60 (0.05)[4.88] [-4.47] [1.75] [-19.61] [3.43] [12.29]

RISK -4.19 (1.39) 0.72 (0.01) 0.01 (0.02) 0.17 (0.03) -0.17 (0.03) -0.62 (0.04)[-3.01] [53.18] [0.40] [6.25] [-5.62] [-15.43]

Average 4.17 (1.11) 0.17 (0.01) 0.05 (0.02) 0.21 (0.02) 0.14 (0.02) 0.36 (0.03)[3.74] [14.84] [3.61] [10.96] [5.36] [11.36]

(c) Daily

Intercept Mkt_RF SMB HML RMW CMA

coef se coef se coef se coef se coef se coef set t t t t t

SMB 4.30 (1.05) -0.11 (0.00) 0.04 (0.01) -0.46 (0.01) -0.07 (0.01)[4.08] [-22.79] [4.32] [-39.87] [-4.71]

HML 1.96 (0.90) -0.01 (0.00) 0.03 (0.01) -0.12 (0.01) 0.76 (0.01)[2.19] [-2.12] [4.32] [-11.34] [72.89]

RMW 4.44 (0.74) -0.10 (0.00) -0.23 (0.01) -0.08 (0.01) 0.04 (0.01)[6.04] [-31.03] [-39.87] [-11.34] [4.03]

CMA 2.52 (0.62) -0.10 (0.00) -0.02 (0.01) 0.37 (0.01) 0.03 (0.01)[4.05] [-36.69] [-4.71] [72.89] [4.03]

MOM 8.30 (1.42) -0.07 (0.01) 0.09 (0.01) -0.54 (0.01) 0.18 (0.02) 0.39 (0.02)[5.85] [-10.87] [7.75] [-39.87] [10.94] [20.01]

RISK -5.24 (1.30) 0.84 (0.01) -0.09 (0.01) 0.22 (0.01) -0.14 (0.02) -0.56 (0.02)[-4.05] [143.26] [-8.26] [17.70] [-9.52] [-31.12]

Average 4.46 (1.00) 0.20 (0.00) 0.08 (0.01) 0.21 (0.01) 0.16 (0.01) 0.30 (0.01)[4.38] [41.13] [10.82] [24.35] [12.62] [22.13]

46

Table 3: Power Di�erence between the CS and TS Test

Asymptotic Small SampleCS TS CS-TS CS TS CS-TS

SR = 0.25Base 17.3% 12.2% 5.1% 17.0% 11.8% 5.2%N = 50 14.8% 10.7% 4.1% 14.6% 10.4% 4.2%N = 10, 000 17.9% 12.5% 5.4% 17.6% 12.1% 5.5%σε = 7 17.9% 13.0% 5.0% 17.6% 12.5% 5.1%σε = 0 11.2% 8.8% 2.3% 11.0% 8.6% 2.4%T = 50 8.0% 6.8% 1.2% 7.7% 6.4% 1.3%T = 1, 000 61.8% 41.2% 20.6% 61.6% 40.9% 20.8%

SR = 0.50Base 52.5% 34.3% 18.2% 51.8% 32.9% 18.9%N = 50 44.2% 28.6% 15.6% 43.6% 27.4% 16.1%N = 10, 000 54.5% 35.7% 18.8% 53.8% 34.3% 19.5%σε = 7 54.6% 37.5% 17.1% 53.9% 36.0% 17.9%σε = 0 30.4% 20.8% 9.6% 29.9% 19.9% 9.9%T = 50 17.3% 12.2% 5.1% 16.1% 10.5% 5.6%T = 1, 000 99.5% 93.5% 6.0% 99.5% 93.3% 6.2%

SR = 1.00Base 97.7% 86.0% 11.7% 97.5% 84.4% 13.1%N = 50 94.4% 77.8% 16.5% 94.0% 76.0% 18.1%N = 10, 000 98.2% 87.6% 10.6% 98.0% 86.1% 11.9%σε = 7 98.2% 89.4% 8.8% 98.0% 88.0% 10.1%σε = 0 80.6% 60.8% 19.9% 80.1% 58.8% 21.3%T = 50 50.6% 33.0% 17.6% 47.9% 27.4% 20.5%T = 1, 000 100.0% 100.0% 0.0% 100.0% 100.0% 0.0%

SR = 1.50Base 100.0% 99.6% 0.4% 100.0% 99.4% 0.6%N = 50 100.0% 98.5% 1.5% 100.0% 98.0% 1.9%N = 10, 000 100.0% 99.7% 0.3% 100.0% 99.6% 0.4%σε = 7 100.0% 99.8% 0.2% 100.0% 99.7% 0.3%σε = 0 99.0% 92.2% 6.8% 98.8% 91.0% 7.9%T = 50 84.8% 63.2% 21.6% 82.6% 54.5% 28.1%T = 1, 000 100.0% 100.0% 0.0% 100.0% 100.0% 0.0%

Note: The annual Sharpe ratio of the test factor is calucated as SR = µbσbb·√

12.

47

A Extensions to the Basic Model

A.1 Variable Transaction Costs

So far, we have considered the case where every stock has the same level of transaction costs θ.

In reality, producing a factor portfolio with a given exposure to a particular set of scores varies in

terms of its execution cost. This is in part because the extreme scores may tilt towards smaller,

more volatile, or less liquid securities, which have a higher cost θi than those in the middle. The

change in our investor's weights is slight, as we replace θ with a vector of transaction costs θ.

w∗vtc = diag (θ)−1 Γµ (45)

The stock level weights are simple, and intuitive. Instead of scaling the expected return by

a common transaction cost θ, each stock has its own stock-speci�c transaction cost scalar θi. In

the case of simple transaction costs in Equation 14, the stock-level logic extends to factors. The

exposure to all factor portfolios scales up or down with a single transaction cost parameter. The

factor exposures with variable transaction costs are more complicated. To simplify notation, we

assume again that the factors are de�ned to be orthogonal so that Γ′Γ = NI and the raw and unit

exposures are the same. Also, we suppose for simplicity that all of the factors are structured so

that the mean return µ is positive. This is not necessary, but it serves to simplify the notation and

develop intuition.

eraw (w∗vtc) = eunit (w∗vtc) = Γ′diag (θ)−1 Γµ

= 1N

∑

iγ2aiθi

∑iγai·γbiθi

· · ·∑iγbi·γaiθi

∑iγ2biθi

· · ·...

...

µ(46)

Our investor chooses higher exposure to a factor when its return µa is high relative to its execu-

tion cost. Execution costs are low when extreme scores are negatively correlated with transaction

costs. In other words, exposure is higher on a factor portfolio if the securities with the highest

absolute scores, either positive or negative, so that γ2ai is large, also happen to be easier to trade,

with low θi. There is a second, subtler force in the o�-diagonal terms. Exposure is higher when the

factor of interest has scores that are correlated with other high return µb factors, so that γai · γbi

48

is large, among stocks that are easier to trade, with low θi. This is an echo of the unit portfolio

exposures in the case of a risk neutral investor with constant transaction costs and correlated scores.

Here the scores are by assumption uncorrelated, but incidental positive or negative exposures can

nonetheless arise if there is a conditional positive or negative score correlation in the set of stocks

that are cheaper to trade. This is one reason to prefer a single multi-factor portfolio optimization

to a portfolio of factor portfolios, because it takes into account execution savings across factor

portfolios.

The factor exposures can be further simpli�ed in the case where the scores are also uncorrelated

conditional on transaction costs, so that E (γai · γbi|θ) = 0. That eliminates any spillover from one

factor exposure to another:

eraw (w∗vtc) = eunit (w∗vtc) =

1N

∑iγ2aiθiµa

1N

∑iγ2biθiµb

...

(47)

The conclusion is that our investor uses a simple formula to adjust expected returns for execution

costs, which here are increasing in the correlation of trading costs with extreme scores. This will

shift the tests in Equation 15 and 25, replacing the gross return with the net return above that

adjusts for transaction costs. For the simple case of uncorrelated scores, the cross section test of

statistical signi�cance is unchanged.

Example 5

Consider an example of two factors, using low asset growth and operating pro�tability once again.

Suppose that these two factors are constructed so that they have zero correlation, even conditional

on transaction costs. Also assume that the extreme asset growth stocks, those that are growing very

fast or contracting signi�cantly, are less liquid and more costly to trade. Moreover, assume that

�rms with high operating pro�ts are more liquid with lower cost to trade, so that∑ γ2

biθi>∑ γ2

aiθi,

which is roughly consistent with US data. The optimal exposures to the unit or raw factor portfolios,

given that they are unconditionally uncorrelated, are:

49

eraw (w∗vtc) = eunit (w∗vtc) =

eunit,−∆AA

eunit,OPA

=

1N

∑iγ2aiθiµa

1N

∑iγ2biθiµb

(48)

The optimal portfolio has positive exposure to both the unit low asset growth portfolio and

the unit operating pro�ts portfolio. If they have the same gross factor portfolio return µa = µb,

then unit exposure to the operating pro�ts portfolio is higher because its execution costs are lower,

making its net factor portfolio returns higher.

A.2 Capacity Constraints

So far, we have sidestepped the issue of the capacity of the factor portfolios. Our investor has

constant absolute risk aversion and no constraints on leverage, so there are no natural e�ects of

assets under management, which we label A. A practical way of making capacity relevant is to

add a few changes to the basic set up in Equations 1 and 11. We �rst imagine that returns in

Equation 1 are de�ned relative to a benchmark, so that all of the factors have zero mean. Second,

we suppose that our investor delegates his portfolio decision, while insisting on some �xed level

of gross exposure, a �xed tracking error, or a minimum level of benchmark-adjusted return. For

example, our investor might ask for a dollar neutral portfolio, where, for each dollar of equity, one

dollar must be invested long and one dollar must be invested short. Or, as we do in Appendix B,

we solve a typical case of �xed tracking error w′var (r) w = σ2T . We proceed here with a further

simpli�ed, risk neutral case where the exposure constraint is expressed as w′w = 1.

max (µΓ)′w − A

2w′diag (θ) w s.t. w′w =1 (49)

Our investor's new optimal weights a slight variation of the unconstrained version:

w∗aum = (A · diag (θ) + 2C (A) I)−1 Γµ (50)

The Lagrange multiplier is C (A), which satis�es (Γµ)′ (Adiag (θ) + 2C (A) I)−2 (Γµ) = 1 and

dCdA < 0 as we show in Appendix B. Again, the weight on any given security is limited by its security

speci�c transaction costs in the �rst term in parenthesis. Now, there is also a second consideration.

We explore these e�ects of assets under management on the factor portfolio exposures. To simplify

50

notation, we assume again that the factors are de�ned to be orthogonal so that Γ′Γ = NI.

eraw (w∗aum) = eunit (w∗aum) = 1NΓ′ (A · diag (θ) + C (A) I)−1 Γµ

= 1N

∑

iγ2ai

Aθi+C(A)

∑i

γai·γbiAθi+C(A) · · ·∑

iγbi·γai

Aθi+C(A)

∑i

γ2bi

Aθi+C(A) · · ·...

...

µ(51)

Our investor again chooses higher exposure to a factor when its return µa is high relative to

its execution cost. Execution costs are low when extreme scores γ2aare negatively correlated with

execution costs. But execution costs themselves are now a function of assets under management,

with θi replaced by Aθi+C (A). As before, the factor exposures can be further simpli�ed in the case

where the scores are also uncorrelated conditional on transaction costs, , so that E (γai · γbi|θ) = 0:

eraw (w∗aum) = eunit (w∗aum) =

1N

∑i

γ2ai

Aθi+C(A)µa

1N

∑i

γ2bi

Aθi+C(A)µb...

(52)

The key insight here is that dCdA < 0 and C does not vary across stocks i. So, for very small assets

under management A, the factor allocation is directly proportional to return µ as in Equation 14.

The gross exposure constraint binds so quickly that transaction costs have no e�ect. As assets rise,

the importance of transaction costs θi increases to the point that the allocations are proportional to

those in Equation 47. The conclusion is that our investor uses a formula to adjust expected returns

for execution costs, but this formula is highly dependent on assets under management, so the exact

adjustment of the tests in Equation 15 and 25, which replace gross returns with net returns are

context speci�c. An anomaly for one investor may not be an economically meaningful anomaly for

another, because its execution costs are too high at the relevant level of assets A. Again, for the

simple case of uncorrelated scores, the cross section test of statistical signi�cance is unchanged.

A.3 Dynamic Trading

So far, we have considered a static trading decision. A fully dynamic optimization with factor scores

Γ that vary through time is beyond the scope of this paper. We analyze a very simple case where

our risk neutral investor trades over a �nite number of periods T. The assumption of risk neutrality

51

is somewhat awkward, because our investor will simply accumulate positions over time, with the

per period accumulation limited by transaction costs. But, this situation still provides the intuition

that the investor in the �rst period will need to consider the returns not just in the next period

but in subsequent periods, with a discount rate δ. More persistent factors deserve greater weight,

because they will generate returns over multiple future holding periods.

maxT∑t=0

δt(

(µΓt)′wt −

1

2(wt −wt−1)

′ diag (θ) (wt −wt−1)

)(53)

For simplicity, we imagine that the factor payo�s and transaction costs are constant through

time, but the factor scores vary according to an autoregressive process:

Γt = Γt−1diag(ρK×1

)K×K + Λt (54)

This is straightforward to solve by backward induction, with the caveat that without a budget

constraint or risk aversion the weights accumulate through time. Garleanu and Pedersen (2013) also

arrive at a similarly interpretable solution with per period risk aversion by assuming that transaction

costs are proportional to the stock level covariance matrix.

w0 = diag (θ)−1 Γ0

(T∑s=0

(δdiag (ρ))s)µ (55)

Again, we are interested in the resulting factor exposures of our investor's initial portfolio, and

we make the same simplifying assumption that Γ′Γ = NI so that the raw and unit exposures are

the same.

eraw (w∗dt) = eunit (w∗dt) = Γ′diag (θ)−1 Γ(∑T

s=0 (δdiag (ρ))s)µ

= 1N

∑

iγ2aiθi

∑iγai·γbiθi

· · ·∑iγbi·γaiθi

∑iγ2biθi

· · ·...

...

11−δρaµa

11−δρbµb

...

(56)

Our investor chooses higher exposure to a factor when the full present value of its future returns

11−δρaµa is high relative to its execution cost. Execution costs are low as before, but now the bene�ts

of trade extend beyond one period. The factor exposures can again be further simpli�ed in the case

where the scores are also uncorrelated conditional on transaction costs, so that E (γai · γbi|θ) = 0.

52

That eliminates any spillover from one factor exposure to another:

eraw (w∗vtc) = eunit (w∗vtc) =

1N

∑iγ2aiθi

11−δρaµa

1N

∑iγ2biθi

11−δρbµb...

(57)

As an aside, the exposure of persistent factors increases through time, because the portfolio

inherits the e�ects of past decisions. Nonetheless, the relevant information for our investor on a

forward-looking is the execution-cost-adjusted and persistence-adjusted mean returns in Equation

57. Layering on capacity constraints does not produce nicely interpretable exposures, but the e�ect

of combining capacity constraints with dynamic trading is to lower the exposure of costly-to-trade

factors in particular when assets under management are high, and when these factors are less

persistent. As in the previous two subsections, for the simple case of uncorrelated scores, the cross

section test of statistical signi�cance is unchanged.

Example 6

Consider an example of two factors, using operating pro�tability as before and using high frequency

reversal instead of low asset growth. Jegadeesh and Titman (1993) among others have observed

that the �rms with the highest returns in the previous month have lower average returns in the

month that follows. So, a stock with a high trailing one-month return might then have a low high

frequency reversal score γa,i. Again, suppose that these two factors are constructed so that they

have zero correlation with one another. For this particular example, we assume that the sets of

stocks with high and low operating pro�tability are very persistent, so that there is a monthly mean

persistence of ρa = 0.98 in factor returns on some initial set of scores γb. And, we assume that

high frequency reversal scores are essentially uncorrelated through time, with monthly persistence

of ρb = 0. This is roughly consistent with US data. Finally, we use a monthly discount rate δ = 0.9,

meaning that the investor's portfolio might fully turnover about once per year. The dynamic target,

optimal exposures to the unit or raw factor portfolios are:

eraw (w∗vtc) = eunit (w∗vtc) =

eunit,rt−1

eunit,OPA

=

1N

∑iγ2aiθi

(1

1−0.9·0.98µa

)1N

∑iγ2aiθiµb

(58)

53

The optimal portfolio has positive exposure to both the unit high frequency reversal portfolio

and the unit operating pro�ts portfolio. If they have the same gross, per period factor portfolio

return µa = µb, then exposure to operating pro�ts is approximately 8.5 times higher because of its

much higher persistence.

B Derivations

B.1 Derivation of Equation 25

Here, we would like to prove that Equation 25 is equivalent to Equation 24:

∑k 6=a

σakeunit,k (w∗ra) = µa (59)

First, we substitute in the optimal unit exposure from Equation 22 and rearrange the terms,

putting µaon the left-hand side.

µa = 2λ (σabeb + σacec + . . . ) (60)

Second, we note that Equation 25 can be rewritten as follows:

µa =∑k 6=a

βakµk = 0⇒ µa =∑k 6=a

βakµk =∑k 6=a

βak2λ

(∑l=b

σklel

)(61)

Now, it remains to show that the right hand side of these two previous equations are the same.

This will be true if the following holds for all factors l not equal to the test factor a:

∀l, σal =∑k 6=a

βakσkl (62)

This identity is simply the identity that the covariance of the sum is equal to the sum of

covariances:

cov (fa, fl) = cov

(∑k=b

βakfk, fl

)=∑k=b

βakcov (fk, fl) (63)

54

B.2 Solution for Equation 49 with a Budget Constraint

Here, we would like to derive the optimal weights for an investor who faces variable transaction

costs with variable assets under management. The general optimization problem is repeated from

Equation 49 above:

max (µΓ)′w − A

2w′diag (θ) w s.t. w′w = 1 (64)

We solve the Lagrangian, where we use the parameter C as the Lagrange multiplier, noting

along the way that C is a function of assets under management A,

L = (µΓ)′w − A

2w′diag (θ) w − C (A)

(w′w − 1

)(65)

by taking the �rst order condition with respect to portfolio weights:

∂L∂w

= 0 (66)

⇒ µΓ− (A · diag (θ) + 2C (A) I) w = 0 (67)

This gives us a solution forw:

w = (A · diag (θ) + 2C (A) I)−1 µΓ (68)

B.3 Derivation of∂C(A)∂A

< 0 in Equation 50

Here, we would like to show that the Lagrange multiplier, C (A), from the previous section, and in

Equation 50 is decreasing in assets under management, A. The FOC can rewritten as follows:

(Γµ)′ (A · diag (θ) + 2C (A) I)−2 (Γµ) = 1 (69)

⇒ µR′ (A · diag (θ) + 2C (A) I)−2µR = 1 (70)

Because A · diag (θ) + 2C (A) I is an diagonal matrix, its inverse is still an diagonal matrix with

inverses of corresponding diagonal elements. We can then express the squared inverse as follows:

55

(A · diag (θ) + 2C (A) I)−2 =

1

(Aθ1+2C(A))2

1(Aθ2+2C(A))2

. . .

(71)

Substituting for the squared inverse, simplifying, performing matrix multiplication, gives us the

following �rst order condition:

µR,1

µR,2...

′

1(Aθ1+2C(A))2

1(Aθ2+2C(A))2

. . .

µR,1

µR,2...

′

= 1 (72)

⇒[

µR,1

(Aθ1+2C(A))2µR,2

(Aθ2+2C(A))2· · ·

]µR,1

µR,2...

′

= 1 (73)

⇒N∑i=1

µ2R,i

(Aθi + 2C (A))2= 1 (74)

Moving all terms to left hand side of the equation, we can now label the new FOC as F (A):

F (A) =N∑i=1

µ2R,i

(Aθi + 2C (A))2− 1 = 0 (75)

Finally, we apply the implicit function theorem and express total derivative in terms of partial

derivatives,

dF

dA= 0 (76)

⇒ ∂F

∂A+

∂F

∂C (A)

∂C (A)

∂A= 0 (77)

and we rearrange terms, solving for ∂C(A)∂A :

∂C (A)

∂A= − ∂F/∂A

∂F/∂C (A)(78)

Because the numerator and denominator are both negative,

56

∂F

∂A= −2

∑i

θiµ2R,i

(Aθi + 2C (A))3< 0 (79)

∂F

∂C (A)= −2

∑i

µ2R,i

(Aθi + 2C (A))3< 0 (80)

we have proven that ∂C(A)∂A < 0.

B.4 Derivation of Equation 35 for the Cross Section Test

Here, we derive the standard error formula, Equation 35, for the cross section test. For the cross

section test, we start by running a cross-sectional regression at each time t:

rt ∼ Γtft + εt (81)

where

Γ′ =

1 . . . 1

γa1 . . . γaN

γb1 . . . γbN...

......

= [γ1, · · · ,γN ] (82)

We obtain the regression coe�cient at time t:

ft =(Γ′tΓt

)−1Γ′trt (83)

and we introduce the notation δ to refer to the columns of (Γ′Γ)−1:

Γ′Γ = N

1 γa γb · · ·

γ2a γaγb · · ·

γ2b · · ·. . .

(84)

(Γ′Γ

)−1=

1

N

[δ1 δa δb δc · · ·

](85)

We �rst rewrite the regression coe�cient ft by substituting for rt with its de�nition:

ft = ft +(Γ′Γ

)−1Γ′εt (86)

The term Γ′εt can be rewritten as:

57

Γ′εt =

∑i εit∑i γaiεit∑i γbiεit

...

=∑i

εit

γaiεit

γbiεit...

=∑i

γiεit (87)

We then substitute this into the formula for ft and obtain the regression coe�cient for a test

factor which we label arbitrarily as b as follows:

fb,t = fb,t +1

Nδ′b∑i

γiεit (88)

Now, we note that the test statistics for the cross section test is the time-series means of the

regression coe�cients is:

µ =1

T

∑t

ft (89)

This means that the standard error of the test statistic for test factor b is:

se (µb) =1√T

√σ2bb +

1

N2σ2∑i

(δ′bγi

)2(90)

B.5 GMM Derivation of Equation 40 for the Time Series Test

Here, we derive the standard error formula, Equation 40, for the time series test. We start by noting

that the return generating process is as follows:

rt = Γtft + εt (91)

Next, we run a Jensen's alpha test using an arbitrary test factor b by regressing its return on

existing factors:

fb,t ∼ αb + β′bf−b,t + εt (92)

If the test factor's returns are orthogonal to all other factor returns, then αb = µb, the mean

return of the test factor portfolio from the return generating process. If the test factor's returns

are not orthogonal and are instead fully spanned by the other factor returns, then αb = 0. Now, we

58

note that the test statistic for the time series test is the intercept αb from this regression, where we

substitute in population estimates for the parameters in the return generating process:

αb = fb − β′bf−b = µb − β

′bµ−b (93)

We then run a GMM estimation using the OLS moments to estimate coe�cients, following the

standard approach in Cochrane (2009, Ch 12):

gT (b) = ET

εt

fa,tεt

fc,tεt...

= 0 (94)

where we de�ne b as the vector of parameters to be estimated:

b =[α, βba, βbc · · ·

]′(95)

so that α can be written as follows:

α = fb −

βba

βbc...

′

·

fa

fc...

(96)

Note that the GMM estimates here are the same as the OLS regression coe�cients, and the full

asymptotic joint distribution of the GMM estimates is as follows:√T (b− b)→ N

(0, (AD)−1ASA′(AD)−1′

)(97)

where the matrices A, D, and S are de�ned next. Because AgT (b) = 0, A = I and it drops out.

Matrix D, according to the GMM formula, is2:

D ≡ ∂gT (b)

∂b= −Φ = −

1 f−b′

f−b Σ−b + f−b · f−b′

(98)

2From Cochrane (2012): precisely, D is de�ned as the population moment in the �rst equality, which we estimatein sample by the second equality

59

where we de�ne f−b and Σ−b as follows:

f−b = (fa, fc, · · · )′ (99)

Σ−b =1

T

∑t

(f−b,t − f−b

) (f−b,t − f−b

)′(100)

The third matrix S is de�ned as:

S ≡∞∑

j=−∞E

εtεt−j εtεt−jfa,t−j εtεt−jfc,t−j · · ·

εtfa,tεt−j εtfa,tεt−jfa,t−j εtfa,tεt−jfc,t−j · · ·

εtfc,tεt−j εtfc,tεt−jfa,t εtfc,tεt−jfc,t−j · · ·...

......

. . .

(101)

Additionally, we assume that the residual terms are uncorrelated over time, are homoskedastic,

and the factors other than the test factor b is orthogonal to the residuals in the return generating

process for factor b. This means that the matrix S can simpli�ed to:

S ≡ E

ε2t ε2t fa,t ε2t fc,t · · ·

ε2t f2a,t ε2t fa,tfc,t · · ·

ε2t f2c,t · · ·

= ΦE[ε2t ] (102)

Note when the test factor portfolio return fb are orthogonal to the other factor returns, E[ε2t ] =

σ2bb. Now, we can substitute the matrices A, D and S into the asymptotic distribution and obtain

the variance of the GMM estimate b:

V ar(b) =1

TD−1SD−1′ =

1

TΦ−1Φσ2εΦ

−1 =1

TΦ−1σ2ε (103)

To obtain the variance of the test statistic V ar(α), we need to calculate the top left corner of

the matrix V ar(b). First, we calculate the top left corner of Φ−1. To do so, we perform a matrix

inversion, which takes the following form:

Φ−1 ≡

C−11 · · ·

· · · C−12

(104)

where, in our case, the upper left block is simply a scalar, though we continue to refer to it as the

matrix C1:

C1 = 1− f−b′(Σ−b + f−b · f−b

′)−1

f−b (105)

The inverse of(Σ−b + f−b · f−b

′)−1

can be rewritten as:

60

Σ−1−b − Σ

−1−bf−b

(1 + f−b

′Σ−1−bf−b

)−1f−b′Σ−1−b (106)

which we can then substitute into the formula for C1, simplify, and invert:

C1 = 1−f−b′Σ−1−bf−b

1 + f−b′Σ−1−bf−b

(107)

=1

1 + f−b′Σ−1−bf−b

(108)

⇒ C−11 = 1 + f−b′Σ−1−bf−b (109)

= 1 + µ′−bΣ−1−bµ−b (110)

Therefore, the standard error of the test statistics αb is:

se (αb) =

√1

TC−11 σ2ε (111)

=1√T

√σ2ε(1 + µ′−bΣ

−1−bµ−b

)(112)

From Appendix B.6, we derive the variance of the e�cient portfolio for factor b is σ2bb +

1N2σ

2∑

i

(δ′bγi

)2. Therefore, by decomposition of the total variance, we get:

σ2bb +1

N2σ2∑i

(δ′bγi

)2= β′bΣ−bβb + σ2ε (113)

⇒ σ2ε =(1−R2

)(σ2bb +

1

N2σ2∑i

(δ′bγi

)2)(114)

where R2 corresponds the regression in Equation 92.

Plug it back in the Equation for se (αb) and get:

se (αb) = 1√T

√(1−R2)

(σ2bb + 1

N2σ2∑

i

(δ′bγi

)2)(1 + SR2) (115)

B.6 Using an E�cient Factor Portfolio in Equation 40

Here, we derive the form of Equation 40 using an e�cient portfolio for a test factor b. This is

essentially the same as forming a portfolio for test factor b using a cross section regression, as

follows. We use optimization to construct a pure test factor portfolio P that is dollar neutral,

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delivers unit exposure to the test factor of interest, zero exposure to all other factors, and otherwise

minimizes idiosyncratic risk.

Let the weight in portfolio P be w′ = [w1, w2, . . . , wN ]. The portfolio has the following return

properties:

rPt = fbt + εPt (116)

µP = µb (117)

σ2P = σ2bb + σ2Pε (118)

We then minimize idiosyncratic risk:

minσ2Pε (119)

subject to constraints of dollar neutrality, unit exposure to b and zero exposure to all other factors:

∑iwi = 0∑iwiγai = 0∑iwiγbi = 1∑iwiγci = 0

...

(120)

The solution has the same form as the unit exposure portfolios Qunit = Γ (Γ′Γ)−1. And so, the

portfolio variance can be written as:

σ2P = σ2bb + σ2Pε = σ2bb +1

N2σ2∑i

(δ′bγi

)2(121)

which is the same as the variance of the individual cross-sectional regression coe�cient.

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